Toshikazu Ikeda

Possibilities for, and Obstacles to Teaching Applications and Modelling in the Lower Secondary Levels

Reports on teaching applications and modelling in eight countries dealt with similar arguments regarding obstacles, each with a different emphasis. The following four obstacles are seem to be common across eight countries in which applications and modelling are located within the national curriculum; “Teachers’ perceptions of mathematics”, “Teachers’ understanding of modelling”, “A lack of adequate textbooks and curricular modelling tasks”, “A lack of adequate assessment, and of modelling tasks in central examinations”.

Pedagogical Reflections on the Role of Modelling in Mathematics Instruction

From a broad perspective, there are two categories of pedagogical aims of modelling. The first is where modelling itself is treated as an objective. The second is mathematical knowledge construction, where modelling is treated as a means to an end. For the first aim three key questions are considered: (1) How can the teacher set an appropriate situation so that students realise the necessity of solving a real-world problem? (2) How can the teacher assist students’ abstraction processes? (3) How can the teacher show students the necessity of controlling various assumptions? For the second aim, three principles are suggested: (1) expanding and clarifying real-world situations satisfying a developed original model, (2) expanding and integrating mathematical knowledge by setting up a concrete situation so that students can consider it, and (3) refining and clarifying the developed mathematical methods by treating instances of the same contexts repeatedly.

Mathematical Applications and Modelling in the Teaching and Learning of Mathematics

Co-chairs: Jill Brown and Toshikazu Ikeda; Team Members: Sung Sook Kim, (Korea), Nicholas Mousoulides (Cyprus), Jussara de Loiola Araujo (Brazil); Liaison IPC member: Morten Blomhoej (Denmark)

Making Connections Between Modelling and Constructing Mathematics Knowledge: An Historical Perspective

This study will look at a surprising resolution of the tension that arises in trying to strike a balance between modelling and pure mathematics by examining Japanese textbooks for the junior high school nearly 70 years ago. Three characteristics are found: (1) two distinct roles –first as objects to mathematize in order to solve real world problems and second as evidence by which to test the validity of mathematical concepts; (2) repeated instances of the same contexts through which new phases of mathematization could be developed; and (3) a series of real world questions focussed on the reason for solving a real world problem.

Applying PISA Ideas to Classroom Teaching of Mathematical Modelling

This chapter argues that the Mathematics Framework of PISA provides a meaningful guide for practical classroom teaching focused on mathematical modelling. The chapter discusses in detail how the Framework can provide guidance on choosing problem situations that interest students and also guide teaching students to appreciate the ways in which mathematics is used by society. In order to supplement the teaching of modelling through holistic problems involving all aspects of the modelling cycle, the chapter recommends the use of PISA-type problems to foster specific modelling competencies such as selecting variables and generating relationships. Advice on how this can be done is backed up by reports of experimental teaching. Finally, the effects of PISA in Japan are briefly discussed.

Reconsidering the Roles and Characteristics of Models in Mathematics Education

A model is generally assumed to be built by translating a real world problem into a mathematical representation. We attempt to re-construct this interpretation and point to at least two distinct meanings: (Role 1) models as hypothetical working spaces, and (Role 2) models as physical/mental entities for comparing and contrasting. This leads us to draw attention to four different perspectives of modelling: (a) where modelling is interpreted as interactive translations among plural worlds not between two fixed worlds; (b) where models have the potential to incorporate scenarios beyond the initial problem situation; (c) where the mathematical world is used as a source of mental entities for comparing and contrasting; (d) where modelling competency means knowing how to balance between these different roles. Perspectives (a) and (d) are concerned with Role 1 and (b), (c) and (d) are concerned with Role 2.

Comparing an Analytical Approach and a Constructive Approach to Modelling

This study investigates the effects when an analytical approach is used in modelling as compared to a constructive approach. In the first approach, students were given a simple mathematical representation of the situation being modeled, while in the second, students were given some key questions to guide them in creating a suitable representation. Although the results were not totally in favor of a constructive approach, it does appear that students using an analytical approach tended to focus too much on the given mathematical representation without paying sufficient attention to the assumptions and limiting conditions implicit in the situation.

Perspectives of cross-curricular activity in Japanese mathematics education

Historically, we can find cross-curricular activities (“life unit study”) after World War II in Japan. When we consider the arguments of “life unit study”, the following three approaches will serve as reference: the approach orientated to the independent aspect, the approach orientated to the societal aspect, the approach orientated to the scientific aspect. By reflecting the critique of “life unit study”, it is suggested that a balanced approach should be carried out. In other words, teachers had better focus on one aspect according to the aim of classroom teaching, and balance three approaches on the whole.

Modelling as Interactive Translations Among Plural Worlds: Experimental Teaching Using the Night-Time Problem

This case study examines the advantages in interpreting students’ modelling from the perspective of interactive translations among plural worlds. This perspective has greater pedagogical potential than a simplified perspective which treats mathematical modelling as involving transitions only between two fixed worlds – a real world and a mathematical world. Experimental lessons with the Night-Time Problem for Japanese 10th grade students were held over a period of 100 min, using a structured investigation. As a result, the following two advantages are exemplified. The first enables a teacher to direct attention to intermediate models which can help students build further abstract models, and the second focusses attention on meaningful contradictions to help students to verify/critique/modify their original models.